Group #36520176 — subgroups

Subgroups — #36520176

Order 22784.

712 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 323 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028883	C₂	1	3	yes	maximal
2	#26018555	C₂	1	3	yes	maximal, complemented
3	#26018553	C₂	1	6	yes	maximal, complemented
4	#26018551	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
7	#19151653	C₄	1	12	yes	—
8	#19151649	C₂²	1	6	yes	—
9	#19151649	C₂²	1	6	yes	—
10	#19151621	C₄	1	6	yes	—
11	#19151561	C₄	1	6	yes	—
12	C₄×#13083558	C₂²	1	3	yes	—
13	#19151320	C₂²	1	6	yes	complemented
14	#19151319	C₂²	1	3	yes	complemented
15	C₈×#10623494	C₂²	1	3	yes	complemented
16	C₄×M₁₆×C₈₉	C₂²	1	3	yes	—
17	#322×C₈₉	C₂²	1	3	yes	complemented
18	C₈²×C₈₉	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₄×D₈₉	C₂×C₄	1	3	yes	—
20	C₂×#13083558	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₈₉	C₂×C₄	1	3	yes	—
22	C₄²×D₈₉	C₂×C₄	1	6	yes	—
23	C₄²×D₈₉	C₂³	1	2	yes	—
24	#15667941	C₂×C₄	1	12	yes	—
25	#15667941	C₂×C₄	1	12	yes	—
26	#15667924	C₂×C₄	1	12	yes	—
27	#15667924	C₂×C₄	1	12	yes	—
28	#15667923	C₂×C₄	1	6	yes	—
29	C₄×#10623494	C₂³	1	6	yes	—
30	C₂×M₁₆×C₈₉	C₂×C₄	1	6	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#97×C₈₉	C₂×C₄	1	6	yes	—
33	C₄×C₈×C₈₉	C₂³	1	3	yes	—
34	C₄×C₈×C₈₉	C₂³	1	3	yes	—
35	C₂³×D₈₉	C₄²	1	1	yes	characteristic
36	C₂²×Dic₈₉	C₄²	1	3	yes	—
37	C₂×C₄×D₈₉	—	4	6	no	—
38	C₂×C₄×D₈₉	C₄²	1	6	yes	—
39	C₂×C₄×D₈₉	C₂²×C₄	1	6	yes	—
40	#13083558	—	4	6	no	—
41	C₂×#10623494	C₂²×C₄	1	12	yes	—
42	C₄×Dic₈₉	C₄²	1	6	yes	—
43	C₄×Dic₈₉	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
45	M₁₆×C₈₉	—	4	6	no	—
46	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
47	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
48	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
49	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
51	C₂²×D₈₉	—	4	3	no	—
52	C₂²×D₈₉	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₈₉	—	2	6	no	—
54	C₂×C₄×C₈₉	C₂×C₄²	1	3	yes	—
55	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
56	C₂×Dic₈₉	—	4	3	no	—
57	C₂×Dic₈₉	C₂×C₄²	1	6	yes	—
58	C₄×D₈₉	—	2	12	no	—
59	C₄×D₈₉	—	4	6	no	—
60	C₈×C₈₉	—	2	6	no	—
61	C₈×C₈₉	—	2	6	no	—
62	#10623494	—	2	12	no	—
63	C₂²×C₈₉	—	2	3	no	—
64	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
65	C₂×D₈₉	—	2	6	no	—
66	C₂×D₈₉	—	4	4	no	—
67	C₄×C₈₉	—	2	3	no	—
68	C₄×C₈₉	#568	1	6	yes	—
69	Dic₈₉	—	2	6	no	—
70	#17339	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	—	4	1	no	—
72	C₂×C₈₉	#2873	1	3	yes	—
73	D₈₉	—	4	2	no	—
74	#2920	—	89	3	no	—
75	#1455	—	89	3	no	—
76	#1455	D₈₉	1	1	yes	characteristic, complemented
77	#1454	—	89	6	no	—
78	#1454	—	89	2	no	—
79	C₈₉	#17339	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₄²	—	89	1	no	—
81	#432	—	89	12	no	—
82	#431	—	89	6	no	—
83	#431	—	89	6	no	—
84	#423	—	89	6	no	—
85	#407	—	89	6	no	—
86	C₄×M₁₆	—	89	3	no	—
87	C₄×M₁₆	C₂×D₈₉	1	3	yes	—
88	#322	—	89	3	no	—
89	#322	—	89	6	no	—
90	#322	C₂×D₈₉	1	3	yes	complemented
91	C₈²	—	89	3	no	—
92	C₈²	C₂×D₈₉	1	1	yes	characteristic, complemented
93	C₂³×C₄	—	89	3	no	—
94	C₂×M₁₆	—	89	6	no	—
95	C₂×M₁₆	C₄×D₈₉	1	6	yes	—
96	C₂×C₄²	—	89	3	no	—
97	C₂×C₄²	—	89	6	no	—
98	C₂×C₄²	—	89	2	no	—
99	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
100	C₄⋊C₈	—	89	12	no	—
101	C₄⋊C₈	—	89	12	no	—
102	C₂²⋊C₈	—	89	12	no	—
103	C₂²⋊C₈	—	89	12	no	—
104	#97	—	89	6	no	—
105	#97	C₄×D₈₉	1	6	yes	—
106	C₄×C₈	—	89	6	no	—
107	C₄×C₈	C₂²×D₈₉	1	3	yes	—
108	C₄×C₈	C₂²×D₈₉	1	3	yes	—
109	C₂⁴	—	89	1	no	—
110	C₂²×C₄	—	89	3	no	—
111	C₂²×C₄	—	89	6	no	—
112	C₂²×C₄	—	89	6	no	—
113	C₂²×C₄	—	356	6	no	—
114	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
115	M₁₆	—	4	6	no	—
116	M₁₆	—	356	6	no	—
117	C₂×C₈	—	89	12	no	—
118	C₂×C₈	C₂×C₄×D₈₉	1	6	yes	—
119	C₂×C₈	C₂×C₄×D₈₉	1	6	yes	—
120	C₄²	—	89	6	no	—
121	C₄²	—	89	6	no	—
122	C₄²	C₂×C₄×D₈₉	1	3	yes	—
123	C₄²	C₂³×D₈₉	1	1	yes	characteristic
124	C₂³	—	89	2	no	—
125	C₂³	—	356	3	no	—
126	C₂³	C₄²×D₈₉	1	1	yes	characteristic
127	C₂×C₄	—	2	6	no	—
128	C₂×C₄	—	89	6	no	—
129	C₂×C₄	—	356	6	no	—
130	C₂×C₄	—	356	3	no	—
131	C₂×C₄	—	178	12	no	—
132	C₂×C₄	C₄²×D₈₉	1	3	yes	—
133	C₂×C₄	C₂²×C₄×D₈₉	1	3	yes	—
134	C₈	—	2	6	no	—
135	C₈	—	2	6	no	—
136	C₈	—	178	12	no	—
137	C₂²	—	2	3	no	—
138	C₂²	—	356	4	no	—
139	C₂²	—	178	6	no	—
140	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
141	C₄	—	2	3	no	—
142	C₄	—	178	6	no	—
143	C₄	#19152364	1	6	yes	—
144	C₂	—	4	1	no	—
145	C₂	—	356	2	no	—
146	C₂	#26028669	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26028883

C₂

yes

maximal

#26018555

C₂

yes

maximal, complemented

#26018553

C₂

yes

maximal, complemented

#26018551

C₂

yes

maximal, complemented

C₈₉×#1455

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₈₉

C₂²

yes

characteristic

#19151653

C₄

yes

—

#19151649

C₂²

yes

—

#19151649

C₂²

yes

—

#19151621

C₄

yes

—

#19151561

C₄

yes

—

C₄×#13083558

C₂²

yes

—

#19151320

C₂²

yes

complemented

#19151319

C₂²

yes

complemented

C₈×#10623494

C₂²

yes

complemented

C₄×M₁₆×C₈₉

C₂²

yes

—

#322×C₈₉

C₂²

yes

complemented

C₈²×C₈₉

C₂²

yes

characteristic, complemented

C₂²×C₄×D₈₉

C₂×C₄

yes

—

C₂×#13083558

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

C₄²×D₈₉

C₂×C₄

yes

—

C₄²×D₈₉

C₂³

yes

—

#15667941

C₂×C₄

yes

—

#15667941

C₂×C₄

yes

—

#15667924

C₂×C₄

yes

—

#15667924

C₂×C₄

yes

—

#15667923

C₂×C₄

yes

—

C₄×#10623494

C₂³

yes

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#97×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂³×D₈₉

C₄²

yes

characteristic

C₂²×Dic₈₉

C₄²

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

C₄²

yes

—

C₂×C₄×D₈₉

C₂²×C₄

yes

—

#13083558

—

C₂×#10623494

C₂²×C₄

yes

—

C₄×Dic₈₉

C₄²

yes

—

C₄×Dic₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

—

C₄²×C₈₉

C₂⁴

yes

characteristic

C₂³×C₈₉

C₂×C₄²

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

C₂×C₄²

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄²

yes

—

C₂×C₄×C₈₉

C₂³×C₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×C₄²

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×C₄²

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#568

yes

—

Dic₈₉

—

#17339

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2873

yes

—

D₈₉

—

#2920

—

#1455

—

#1455

D₈₉

yes

characteristic, complemented

#1454

—

#1454

—

C₈₉

#17339

yes

characteristic, complemented, Sylow

C₂²×C₄²

—

—

—

—

—

—

C₄×M₁₆

—

C₄×M₁₆

C₂×D₈₉

yes

—

#322

—

#322

—

#322

C₂×D₈₉

yes

complemented

C₈²

—

C₈²

C₂×D₈₉

yes

characteristic, complemented

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

104

#97

—

105

#97

C₄×D₈₉

yes

—

106

C₄×C₈

—

107

C₄×C₈

C₂²×D₈₉

yes

—

108

C₄×C₈

C₂²×D₈₉

yes

—

109

C₂⁴

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

356

—

114

C₂²×C₄

C₂×C₄×D₈₉

yes

—

115

M₁₆

—

116

M₁₆

—

356

—

117

C₂×C₈

—

118

C₂×C₈

C₂×C₄×D₈₉

yes

—

119

C₂×C₈

C₂×C₄×D₈₉

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×C₄×D₈₉

yes

—

123

C₄²

C₂³×D₈₉

yes

characteristic

124

C₂³

—

125

C₂³

—

356

—

126

C₂³

C₄²×D₈₉

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

356

—

130

C₂×C₄

—

356

—

131

C₂×C₄

—

178

—

132

C₂×C₄

C₄²×D₈₉

yes

—

133

C₂×C₄

C₂²×C₄×D₈₉

yes

—

134

C₈

—

135

C₈

—

136

C₈

—

178

—

137

C₂²

—

138

C₂²

—

356

—

139

C₂²

—

178

—

140

C₂²

C₂×C₄²×D₈₉

yes

characteristic

141

C₄

—

142

C₄

—

178

—

143

C₄

#19152364

yes

—

144

C₂

—

145

C₂

—

356

—

146

C₂

#26028669

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Fitting	C₈₉×#1455	C₂
prime radical	C₂³×D₈₉	C₄²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1455	D₈₉
hypocenter	C₈₉	#17339
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #1455

subgroup

G/H

center, Frattini

C₄²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂²×C₄²

Fitting

C₈₉×#1455

C₂

prime radical

C₂³×D₈₉

C₄²

cosocle

C₄²×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₄² ◁ #1455

N	H ≅ G/N	type
2. #26018555	C₂	semidirect
3. #26018553	C₂	semidirect
4. #26018551	C₂	semidirect
5. C₈₉×#1455	C₂	semidirect
13. #19151320	C₂²	semidirect
14. #19151319	C₂²	semidirect
15. C₈×#10623494	C₂²	semidirect
17. #322×C₈₉	C₂²	semidirect
18. C₈²×C₈₉	C₂²	semidirect
76. #1455	D₈₉	semidirect
79. C₈₉	#17339	semidirect
90. #322	C₂×D₈₉	semidirect
92. C₈²	C₂×D₈₉	semidirect

H ≅ G/N

type

2. #26018555

C₂

semidirect

3. #26018553

C₂

semidirect

4. #26018551

C₂

semidirect

5. C₈₉×#1455

C₂

semidirect

13. #19151320

C₂²

semidirect

14. #19151319

C₂²

semidirect

15. C₈×#10623494

C₂²

semidirect

17. #322×C₈₉

C₂²

semidirect

18. C₈²×C₈₉

C₂²

semidirect

76. #1455

D₈₉

semidirect

79. C₈₉

#17339

semidirect

90. #322

C₂×D₈₉

semidirect

92. C₈²

C₂×D₈₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26028883	2
#26018555	2
#26018553	2
#26018551	2
C₈₉×#1455	2
#17339	89

subgroup

index

C₈₉×#1455

Subgroups — #36520176

Conjugacy classes of subgroups

Complements

Maximal subgroups